Quantitative Reasoning: Algebra Flashcards

If 3x - 7 = 5x + 1, what is the value of 2x + 9?

Answer: 1

First, solve for x in the equation 3x - 7 = 5x + 1. Subtract 3x from both sides to get -7 = 2x + 1. Then, subtract 1 from both sides to get -8 = 2x. Divide by 2 to find x = -4. Now, substitute the value of x into the expression 2x + 9. This gives 2(-4) + 9, which simplifies to -8 + 9 = 1.

A rectangle's length is 5 cm more than its width. If the perimeter of the rectangle is 50 cm, what is the area of the rectangle in square cm?

Answer: 150

Let the width be 'w'. Then the length 'l' is w + 5. The perimeter of a rectangle is 2(l + w). So, 2((w + 5) + w) = 50. Simplifying gives 2(2w + 5) = 50, which leads to 4w + 10 = 50. Solving for w, we get 4w = 40, so w = 10 cm. The length is w + 5 = 10 + 5 = 15 cm. The area is length × width, which is 15 cm × 10 cm = 150 square cm.

If a/b = 4 and a - 2b = 10, which of the following is the value of 'a'?

Answer: 20

From the first equation, a/b = 4, we can express 'a' in terms of 'b': a = 4b. Now, substitute this expression into the second equation: (4b) - 2b = 10. This simplifies to 2b = 10, so b = 5. To find 'a', substitute the value of 'b' back into a = 4b. This gives a = 4(5), so a = 20.

In a school, the ratio of boys to girls is 3:5. If there are a total of 720 students, how many more girls are there than boys?

Answer: 180

The ratio 3:5 means there are 3+5 = 8 parts in total. To find the value of one part, divide the total number of students by the total number of parts: 720 / 8 = 90 students per part. The number of boys is 3 parts × 90 = 270. The number of girls is 5 parts × 90 = 450. The difference is 450 - 270 = 180.

If x² - y² = 48 and x - y = 4, what is the value of x + y?

Answer: 12

The expression x² - y² is a difference of squares, which can be factored into (x - y)(x + y). We are given that x² - y² = 48 and x - y = 4. So, we can substitute these values into the factored equation: 4 * (x + y) = 48. To solve for (x + y), divide both sides by 4: (x + y) = 48 / 4 = 12.

A man's age is currently three times his son's age. In 10 years, the man's age will be twice his son's age. What is the man's current age?

Answer: 30

Let the son's current age be 's' and the man's current age be 'm'. From the problem, we have m = 3s. In 10 years, the son's age will be s+10 and the man's age will be m+10. At that time, m+10 = 2(s+10). Substitute m = 3s into the second equation: (3s)+10 = 2(s+10). This becomes 3s + 10 = 2s + 20. Subtracting 2s from both sides gives s + 10 = 20, so s = 10. The son's current age is 10. The man's current age is m = 3s = 3(10) = 30.